David Bohm added point particles to quantum mechanics; the particles
are guided by the wavefunction. Extra variables like these on top
of the wavefunction are often called 'hidden variables' - although
the name is a little unfortunate since often it's the hidden
variables, e.g. the particle positions, rather than the wavefunction
that we can directly see! The appeal of the Bohmian particle positions
is that they're easy to understand - just plain old particles - but
one less appealing feature is that they can seem tacked-on (although
their dynamics is given by the 'probability current' in quantum
mechanics which is quite pretty).

Why not use only the wavefunction? The main difficulty in interpretation
comes from the wavefunction being defined on a very high-dimensional
space (a 3N dimensional configuration-space if you have N particles)
rather than our everyday space. In the Everett (or many-worlds)
version of quantum mechanics the wavefunction on high-dimensional
space is all there is. This leaves the difficult task of recovering
things happening in ordinary space - David Wallace has written a
lot about this. For me the appeal of Everett is that it 'takes
quantum mechanics seriously', taking just the wavefunction as given
and aiming to derive everything else. It's no simple thing recovering
everyday space though, so to me even if it succeeds it feels less
direct than having extra variables which are very straightforwardly
related to our everyday world.

Can we combine the physical obviousness of Bohm and the theoretical
naturalness of Everett? Leafing through atomic physics or chemistry
books we would think that this 'quantum mechanics' is really about
orbitals, that is single electron wavefunctions (which are defined
on our ordinary space rather than the high-dimensional configuration
space). These books are full of sketches of these orbitals in
hydrogen atoms, water molecules, crystals, metals, ... - and orbitals
are the basis of the actual calculations too. So for me the orbitals
are an obvious candidate for 'things for quantum mechanics to be
about'. The difficulty is that not all (many-particle) wavefunctions
can be expressed as products of single-particle wavefunctions, that
is in terms of orbitals. The obstacle to doing this is precisely
entanglement.

But all is not lost! From any many-particle wavefunction we can
derive some so-called 'natural orbitals' - a family of single-particle
wavefunctions naturally associated with it. These are used a lot
in quantum chemistry but aren't as famous as they might be. The
construction is dead easy and theoretically quite lovely - they are
'just' eigenfunctions of the one-particle reduced density matrix.
For me these natural orbitals are a great thing for quantum mechanics
to be about. They live in our everyday three-dimensional space and
often look just like the orbitals sketched in Physics and Chemistry
books, so for me they have the Bohm-style physical obviousness in
spades. And they are derived from the many-particle wavefunction
by a very natural theoretical construction, the reduced density
matrix, so for me avoid seeming tacked-on and in fact feel rather
natural.

If we do not just calculate with them but believe in them as
physically real, the natural orbitals are hidden variables playing
a similar role to the Bohmian particles. But they are derived
within the existing structure of quantum mechanics, rather than
being added on top of it. For these reasons I like to think of them
as 'hidden hidden variables' - hidden variables which have been
hidden in quantum mechanics the whole time.

I mean to look more into how this works out in detail, but already
the idea is very dear to me. I once spent a holiday in New York
just wandering around thinking about nothing but this during the
day and going to various jazz clubs in the evening - it was a good
few days! Also Vijay Iyer signed a secondhand copy of Poincaré's
Last Essays I'd picked up which was pretty awesome.